Sub-Randers metrics

Abstract

We introduce a new class of sub-Finsler metrics, called sub-Randers metrics, obtained by adding a one-form $\beta \in \Gamma(\mathcal{D}^*)$ to a sub-Riemannian metric $a$ on a bracket-generating distribution $\mathcal{D} \subset TM$. We define a sub-Randers manifold as a triple $(M, \mathcal{D}, F)$, where $M$ is an $n$-dimensional smooth manifold and $F(v) = \sqrt{a(v,v)} + \beta(v)$, the condition $\|\beta\|_a < 1$ ensures positive definiteness and convexity. Explicit equations for sub-Randers normal geodesics are derived, and we show that normal geodesics depend on $\beta$ while abnormal geodesics are determined solely by the bracket-generating distribution $\mathcal{D}$. Furthermore, we show that Zermelo navigation on $\mathcal{D}$ naturally generates sub-Randers normal geodesics. Finally, we prove a Hopf-Rinow type theorem which guarantees the existence of minimizing geodesics despite asymmetry, generalizing classical results to the sub-Randers setting.